nLab Waldhausen S-construction

Idea

The Waldhausen $S_\bullet$-construction is a procedure that produces algebraic K-theory (as an infinite loop space or connective spectrum) from a category or (infinity,1)-category equipped with a Waldhausen structure?.

The construction can take as input any of the following:

• Waldhausen category
• Waldhausen (infinity,1)-category
• stable (infinity,1)-category
• stable derivator
• …

Definition

For Waldhausen categories

Recall from the definition at K-theory that the K-theory spectrum $K(C)$ of the (∞,1)-category $C$ is the diagonal simplicial set on the bisimplicial set $Core(Func(\Delta^n,C))$ of sequences of morphisms in $C$ and equivalences between these (the core of the Segal space induced by $C$).

The Waldhausen S-construction mimics precisely this: for $C'$ a Waldhausen category for every integer $n$ define a simplicial set $S_n C'$ to be the nerve of the category whose

• objects are sequences $0 \hookrightarrow A_{0,1} \hookrightarrow \cdots \hookrightarrow A_{0,n}$ of Waldhausen cofibrations;

  • together with choices of quotients $A_{i j} = A_{0, j}/ A_{0,i}$, i.e. cofibration sequences $A_{0,i} \to A_{0,j} \to A_{i j}$

• morphisms are collections of morphisms $\{A_{i,j} \to B_{i,j}\}$ that commute with all diagrams in sight.

Then one finds that the realization of the bisimplicial set $S_\bullet C'$ with respect to one variable is itself naturally a topological Waldhausen category. Therefore the above construction can be repeated to yield a sequence of topological categories $S^{(n)}_\bullet C'$. The corresponding sequence of thick topological realizations is a spectrum

$$\mathbf{K}(C)_n = |S^{(n)}_\bullet C'|$$

this is the S-construction of the Waldhausen K-theory spectrum of $C'$.

(… roughly at least, need to polish this, see link below meanwhile…)

Related

• algebraic K-theory

• Quillen Q-construction

• K-theory of a stable (infinity,1)-category

• K-theory of a Waldhausen (infinity,1)-category?

• categorified Dold-Kan correspondence

References

For Waldhausen categories

• F. Waldhausen, Algebraic K-theory of spaces, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318-419, pdf.

The Waldhausen S-construction is recalled for instance in section 1 of

• R. W. Thomason, Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, 1990, 247-435.

or in section 1 of

• Paul D. Mitchener, Symmetric Waldhausen K-theory spectra of topological categories (pdf)

For Waldhausen (infinity,1)-categories and stable (infinity,1)-categories

• C. Barwick?, On the algebraic K-theory of higher categories, arXiv:1204.3607.

• Andrew J. Blumberg, David Gepner, Goncalo Tabuada, A universal characterization of higher algebraic K-theory, arXiv:1001.2282.

Other

A combinatorial construction of symmetries due to Nadler has a relation to the S-construction in a special case:

• David Nadler, Cyclic symmetries of $A_n$-quiver representations, arxiv/1306.0070

This short note contains a combinatorial construction of symmetries arising in symplectic geometry (partially wrapped or infinitesimal Fukaya categories), algebraic geometry (derived categories of singularities), and K-theory (Waldhausen’s S-construction). Our specific motivation (in the spirit of expectations of Kontsevich, and to be taken up in general elsewhere) is a combinatorial construction of quantizations of Lagrangian skeleta (equivalent to microlocal sheaves in their many guises). We explain here the one dimensional case of ribbon graphs where the main result of this paper gives an immediate solution.

Interpretation in terms of a categorified Dold-Kan correspondence is discussed in

• Tobias Dyckerhoff, A categorified Dold-Kan correspondence (arXiv:1710.08356)